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Suppose the potential strength $V$ is large enough and $(a_i)_{i\\ge1}$ is bounded. We prove that the spectral generating bands possess properties of bounded distortion, bounded covariation and there exists Gibbs-like measure on the spectrum $\\sigma(H_{\\alpha,V})$. As an application, we prove that $$\\dim_H \\sigma(H_{\\alpha,V})=s_*,\\quad \\bar{\\dim}_B \\sigma(H_{\\alpha,V})=s^*,$$ where"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0909.2301","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-09-12T04:31:00Z","cross_cats_sorted":[],"title_canon_sha256":"1b54c0e5e377f07dd9883959cba99369cfd608f93b97ce5a60e189488deb523f","abstract_canon_sha256":"928875e356020684cc399806b3d94195391889352a1b903eb5595372f185dcdb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:49:24.366858Z","signature_b64":"zB6ieOrXjuocVSl8xdJaZMNy9Kn00FqmM4mADUMdpC0ElFfhPLF2+nY49JFRulMDlQWEX7UTdXvtfrmz/svbDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dddd8d8a4d78b24f96b8ee85fead3a36191aa81d85ee592f11ab4fd724228fa6","last_reissued_at":"2026-07-04T15:49:24.366416Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:49:24.366416Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gibbs-like measure for spectrum of a class of one-dimensional Schr\\\"odinger operator with Sturm potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Qing-Hui Liu, Shen Fan, Zhi-Ying Wen","submitted_at":"2009-09-12T04:31:00Z","abstract_excerpt":"Let $\\alpha\\in(0,1)$ be an irrational, and $[0;a_1,a_2,...]$ the continued fraction expansion of $\\alpha$. Let $H_{\\alpha,V}$ be the one-dimensional Schr\\\"odinger operator with Sturm potential of frequency $\\alpha$. Suppose the potential strength $V$ is large enough and $(a_i)_{i\\ge1}$ is bounded. We prove that the spectral generating bands possess properties of bounded distortion, bounded covariation and there exists Gibbs-like measure on the spectrum $\\sigma(H_{\\alpha,V})$. As an application, we prove that $$\\dim_H \\sigma(H_{\\alpha,V})=s_*,\\quad \\bar{\\dim}_B \\sigma(H_{\\alpha,V})=s^*,$$ where"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.2301","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0909.2301/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"0909.2301","created_at":"2026-07-04T15:49:24.366481+00:00"},{"alias_kind":"arxiv_version","alias_value":"0909.2301v1","created_at":"2026-07-04T15:49:24.366481+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.2301","created_at":"2026-07-04T15:49:24.366481+00:00"},{"alias_kind":"pith_short_12","alias_value":"3XOY3CSNPCZE","created_at":"2026-07-04T15:49:24.366481+00:00"},{"alias_kind":"pith_short_16","alias_value":"3XOY3CSNPCZE7FVY","created_at":"2026-07-04T15:49:24.366481+00:00"},{"alias_kind":"pith_short_8","alias_value":"3XOY3CSN","created_at":"2026-07-04T15:49:24.366481+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3XOY3CSNPCZE7FVY52C75LJ2GY","json":"https://pith.science/pith/3XOY3CSNPCZE7FVY52C75LJ2GY.json","graph_json":"https://pith.science/api/pith-number/3XOY3CSNPCZE7FVY52C75LJ2GY/graph.json","events_json":"https://pith.science/api/pith-number/3XOY3CSNPCZE7FVY52C75LJ2GY/events.json","paper":"https://pith.science/paper/3XOY3CSN"},"agent_actions":{"view_html":"https://pith.science/pith/3XOY3CSNPCZE7FVY52C75LJ2GY","download_json":"https://pith.science/pith/3XOY3CSNPCZE7FVY52C75LJ2GY.json","view_paper":"https://pith.science/paper/3XOY3CSN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0909.2301&json=true","fetch_graph":"https://pith.science/api/pith-number/3XOY3CSNPCZE7FVY52C75LJ2GY/graph.json","fetch_events":"https://pith.science/api/pith-number/3XOY3CSNPCZE7FVY52C75LJ2GY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3XOY3CSNPCZE7FVY52C75LJ2GY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3XOY3CSNPCZE7FVY52C75LJ2GY/action/storage_attestation","attest_author":"https://pith.science/pith/3XOY3CSNPCZE7FVY52C75LJ2GY/action/author_attestation","sign_citation":"https://pith.science/pith/3XOY3CSNPCZE7FVY52C75LJ2GY/action/citation_signature","submit_replication":"https://pith.science/pith/3XOY3CSNPCZE7FVY52C75LJ2GY/action/replication_record"}},"created_at":"2026-07-04T15:49:24.366481+00:00","updated_at":"2026-07-04T15:49:24.366481+00:00"}